By the first theorem of this section,
the subfield generated by the empty set is isomorphic to
Z/pZ for some prime p. By the previous theorem, F
inherits the structure of a vector space over Z/pZ.
If the dimension of this vector space is n, then every element of F can
be uniquely represented as a Z/pZ linear combination
of n given basis vectors, and so the number of elements of
F is pn.